New oscillation solutions of impulsive conformable partial differential equations

Omar Bazighifan; Areej A. Al-moneef; Ali Hasan Ali; Thangaraj Raja; Kamsing Nonlaopon; Taher A. Nofal; Omar Bazighifan; Areej A. Al-moneef; Ali Hasan Ali; Thangaraj Raja; Kamsing Nonlaopon; Taher A. Nofal

doi:10.3934/math.2022892

AIMS Mathematics

2022, Volume 7, Issue 9: 16328-16348. doi: 10.3934/math.2022892

Previous Article Next Article

Research article

New oscillation solutions of impulsive conformable partial differential equations

1.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq
4.
Doctoral School of Mathematical and Computational Sciences, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary
5.
Department of Mathematics, Mahendra College of Engineering, Affiliated to Anna University, Chennai, Salem (Dt), Tamil Nadu, India
6.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
7.
Department of Mathematic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 13 May 2022 Revised: 22 June 2022 Accepted: 27 June 2022 Published: 05 July 2022
MSC : 5B05, 35L70, 35R10, 35R12

Partial fractional differential equations are fundamental in many physical and biological applications, engineering and medicine, in addition to their importance in the development of several mathematical and computer models. This study's main objective is to identify the necessary conditions for the oscillation of impulsive conformable partial differential equation systems with the Robin boundary condition. The important findings of the study are stated and demonstrated with a robust example at the end of the study.
- fractional integrals,
- nonlinear equations,
- conformable partial differential equations,
- impulse,
- oscillation,
- damping term,
- distributed deviating arguments
Citation: Omar Bazighifan, Areej A. Al-moneef, Ali Hasan Ali, Thangaraj Raja, Kamsing Nonlaopon, Taher A. Nofal. New oscillation solutions of impulsive conformable partial differential equations[J]. AIMS Mathematics, 2022, 7(9): 16328-16348. doi: 10.3934/math.2022892

Related Papers:

Abstract

Partial fractional differential equations are fundamental in many physical and biological applications, engineering and medicine, in addition to their importance in the development of several mathematical and computer models. This study's main objective is to identify the necessary conditions for the oscillation of impulsive conformable partial differential equation systems with the Robin boundary condition. The important findings of the study are stated and demonstrated with a robust example at the end of the study.

References

[1]	G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, Inc, New York, 1987.
[2]	O. Bazighifan, M. A. Ragusa, Nonlinear equations of fourth-order with p-Laplacian like operators: Oscillation, methods and applications, P. Am. Math. Soc., 150 (2022), 1009–1020. https://doi.org/10.1090/proc/15794 doi: 10.1090/proc/15794
[3]	O. Moaaz, D. Chalishajar, O. Bazighifan, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7 (2019), 585. https://doi.org/10.3390/math7070585 doi: 10.3390/math7070585
[4]	O. Bazighifan, H. Alotaibi, A. A. A. Mousa, Neutral delay differential equations: Oscillation conditions for the solutions, Symmetry, 13 (2021), 101. https://doi.org/10.3390/sym13010101 doi: 10.3390/sym13010101
[5]	S. Achar, C. Baishya, M. Kaabar, Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives, Math. Methods Appl. Sci., 45 (2022), 4278–4294. https://doi.org/10.1002/mma.8039 doi: 10.1002/mma.8039
[6]	M. Abu-Shady, M. Kaabar, A generalized definition of the fractional derivative with applications, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/9444803
[7]	M. Kaabar, S. Grace, J. Alzabut, A. Özbekler, Z. Siri, On the oscillation of even-order nonlinear differential equations with mixed neutral terms, J. Funct. Spaces, 2021 (2021). https://doi.org/10.1155/2021/4403821
[8]	R. R. Khalil, M. Al. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Com. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[9]	V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Scientific Publishers, Singapore, 1989.
[10]	W. N. Li, On the forced oscillation of solutions for systems of impulsive parabolic differential equations with several delays, J. Comput. Appl. Math., 181 (2005), 46–57. https://doi.org/10.1016/j.cam.2004.11.016 doi: 10.1016/j.cam.2004.11.016
[11]	V. Sadhasivam, T. Raja, T. Kalaimani, Oscillation of system of impulsive neutral partial differential equations with damping term, Int. J. Pure Appl. Math, 115 (2017), 65–81.
[12]	J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
[13]	K. Gopalsamy, B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110–122. https://doi.org/10.1016/0022-247X(89)90232-1 doi: 10.1016/0022-247X(89)90232-1
[14]	L. H. Erbe, H. I. Freedman, X. Z. Liu, J. H. Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Aust. Math. Soc., 32 (1991), 382–400. https://doi.org/10.1017/S033427000000850X doi: 10.1017/S033427000000850X
[15]	S. Hadi, A. H. Ali, Integrable functions of fuzzy cone and $\xi$-fuzzy cone and their application in the fixed point theorem, J. Interdiscip. Math., 25 (2022), 247–258. https://doi.org/10.1080/09720502.2021.1881220 doi: 10.1080/09720502.2021.1881220
[16]	N. Abdul-Hassan, A. H. Ali, C. Park, A new fifth-order iterative method free from second derivative for solving nonlinear equations, J. Appl. Math. Comput., (2021), 1–10. https://doi.org/10.1007/s12190-021-01647-1
[17]	G. E. Chatzarakis, T. Raja, V. Sadhasivam, T. Kalaimani, Oscillation of certain nonlinear impulsive neutral partial differential equations with continuous distributed deviating arguments and a damping term, Dynam. Cont. Dis. Ser. A., 25 (2018), 329–345.
[18]	O. Bazighifan, T. Abdeljawad, Q. M. Al-Mdallal, Differential equations of even-order with p-Laplacian like operators: Qualitative properties of the solutions, Adv. Differ. Equ., 2021 (2021), 96. https://doi.org/10.1186/s13662-021-03254-7 doi: 10.1186/s13662-021-03254-7
[19]	B. Almarri, A. H. Ali, K. S. Al-Ghafri, A. Almutairi, O. Bazighifan, J. Awrejcewicz, Symmetric and Non-Oscillatory Characteristics of the neutral differential equations solutions related to p-Laplacian operators, Symmetry, 14 (2022), 566. https://doi.org/10.3390/sym14030566 doi: 10.3390/sym14030566
[20]	B. Almarri, A. H. Ali, A. M. Lopes, O. Bazighifan, Nonlinear differential equations with distributed delay: Some new oscillatory solutions, Mathematics, 10 (2022), 995. https://doi.org/10.3390/math10060995 doi: 10.3390/math10060995
[21]	B. Almarri, S. Janaki, V. Ganesan, A. H. Ali, K. Nonlaopon, O. Bazighifan, Novel oscillation theorems and symmetric properties of nonlinear delay differential equations of fourth-order with a middle term, Symmetry, 14 (2022), 585. https://doi.org/10.3390/sym14030585 doi: 10.3390/sym14030585
[22]	O. Bazighifan, A. H. Ali, F. Mofarreh, Y. N. Raffoul, Extended approach to the asymptotic behavior and symmetric solutions of advanced differential equations, Symmetry, 14 (2022), 686. https://doi.org/10.3390/sym14040686 doi: 10.3390/sym14040686
[23]	G. E. Chatzarakis, K. Logaarasi, T. Raja, V. Sadhasivam, Interval oscillation criteria for conformable fractional differential equations with impulses, Appl. Math. E-Notes, 19 (2019), 354–369.
[24]	G. E. Chatzarakis, K. Logaarasi, T. Raja, V. Sadhasivam, Interval oscillation criteria for impulsive conformable partial differential equations, Appl. Anal. Discrete Math., 13 (2019), 325–345.
[25]	G. E. Chatzarakis, T. Raja, V. Sadhasivam, On the Oscillation of impulsive vector partial conformable fractional differential equations, J. Critical Rev., 8 (2021), 524–535.
[26]	A. H. Ali, G. A. Meften, O. Bazighifan, M. Iqbal, S. Elaskar, J. Awrejcewicz, A study of continuous dependence and symmetric properties of double diffusive convection: Forchheimer model, Symmetry, 14 (2022), 682. https://doi.org/10.3390/sym14040682 doi: 10.3390/sym14040682
[27]	G. Abed Meften, A. H. Ali, K. S. Al-Ghafri, J. Awrejcewicz, O. Bazighifan, Nonlinear stability and linear instability of Double-Diffusive convection in a rotating with LTNE effects and symmetric properties: Brinkmann-Forchheimer Model, Symmetry, 14 (2022), 565. https://doi.org/10.3390/sym14030565 doi: 10.3390/sym14030565
[28]	B. Qaraad, O. Bazighifan, T. A. Nofal, A. H. Ali, Neutral differential equations with distribution deviating arguments: Oscillation conditions, J. Ocean. Eng. Sci., (2022). https://doi.org/10.1016/j.joes.2022.06.032
[29]	R. P. Agarwal, F. W. Meng, W. N. Li, Oscillation of solutions of systems of neutral type partial functional differential equations, Comput. Math. Appl., 44 (2002), 777–786. https://doi.org/10.1016/s0898-1221(02)00190-6 doi: 10.1016/s0898-1221(02)00190-6
[30]	L. Du, W. Fu, M. Fan, Oscillatory solutions of delay hyperbolic system with distributed deviating arguments, Appl. Math. Comput., 154 (2004), 521–529. https://doi.org/10.1016/S0096-3003(03)00732-X doi: 10.1016/S0096-3003(03)00732-X
[31]	W. N. Li, B. T. Cui, Oscillation for systems of neutral delay hyperbolic differential equations, Indian J. Pure Appl. Math., 31 (2000), 933–948.
[32]	W. N. Li, B. T. Cui, L. Debnath, Oscillation of systems of certain neutral delay parabolic differential equations, J. Appl. Math. Stochastic Anal., 16 (2003), 83–94. https://doi.org/10.1155/S1048953303000066 doi: 10.1155/S1048953303000066
[33]	W. N. Li, L. Debnath, Oscillation of a systems of delay hyperbolic differential equations, Int. J. Appl. Math., 2 (2000), 417–431.
[34]	W. N. Li, F. Meng, On the forced oscillation of systems of neutral parabolic differential equations with deviating arguments, J. Math. Anal. Appl., 288 (2003), 20–27.
[35]	W. X. Lin, Some oscillation theorems for systems of partial equations with deviating arguments, J. Biomath., 18 (2003), 400–407.
[36]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[37]	M. C. Bortolan, G. E. Chatzarakis, T. Kalaimani, T. Raja, V. Sadhasivam, Oscillations in systems of impulsive nonlinear partial differential equations with distributed deviating arguments, Fasc. Math., 62 (2019), 13–33.
[38]	B. T. Cui, Y. Liu, F. Deng, Some oscillation problems for impulsive hyperbolic differential systems with several delays, Appl. Math. Comput., 146 (2003), 667–679. https://doi.org/10.1016/S0096-3003(02)00611-2 doi: 10.1016/S0096-3003(02)00611-2
[39]	Y. K. Li, Oscillation of systems of hyperbolic differential equations with deviating arguments, Acta Math. Sinica, 40 (1997), 100–105.
[40]	O. Moaaz, R. A. El-Nabulsi, O. Bazighifan, Oscillatory behavior of fourth-order differential equations with neutral delay, Symmetry, 12 (2020), 371. https://doi.org/10.3390/sym12030371 doi: 10.3390/sym12030371
[41]	Y. Bolat, T. Raja, K. Logaarasi, V. Sadhasivam, Interval oscillation criteria for impulsive conformable fractional differential equations, Commun. Fac. Sci. Univ. Ank. Ser. A4 Math. Stat., 69 (2020), 815–831.
[42]	A. H. Ali, A. S. Jaber, M. T. Yaseen, M. Rasheed, O. Bazighifan, T. A. Nofal, A comparison of finite difference and finite volume methods with numerical simulations: Burgers equation model, Complexity, 2022 (2022), 1–9. https://doi.org/10.1155/2022/9367638 doi: 10.1155/2022/9367638
[43]	O. Bazighifan, An approach for studying asymptotic properties of solutions of neutral differential equations, Symmetry, 12 (2020), 555. https://doi.org/10.3390/sym12040555 doi: 10.3390/sym12040555
[44]	S. S. Santra, T. Ghosh, O. Bazighifan, Explicit criteria for the oscillation of second-order differential equations with several sub-linear neutral coefficients, Adv Differ Equ., 2020 (2020). https://doi.org/10.1186/s13662-020-03101-1
[45]	N. Yoshida, Oscillation theory of partial differential equations, World Scientific, Singapore, (2008).
[46]	A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889–898. https://doi.org/10.1515/math-2015-0081 doi: 10.1515/math-2015-0081
[47]	G. H. Hardy, J. E. Littlewood, G. P$\acute{o}$lya, Inequalities, Cambridge University Press, Cambridge, UK, (1988).
[48]	O. Bazighifan, P. Kumam, Oscillation theorems for advanced differential equations with p-Laplacian like operators, Mathematics, 8 (2020), 821. https://doi.org/10.3390/math8050821 doi: 10.3390/math8050821

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)